Question

d) Suppose that 𝑋1,𝑋2,...,𝑋𝑛 is a random sample from a distribution with

probability density function:

𝑓(𝑥;𝜃1,𝜃2)={

1

𝜃2

𝑒−𝑥−𝜃1

𝜃2 , 𝑖𝑓 𝑥 ≥𝜃1 𝑎𝑛𝑑−∞<𝜃2 <∞

0, 𝑒𝑙𝑠𝑒 𝑤ℎ𝑒𝑟𝑒

If 𝜃1 =0,

i) show whether or not ∑ 𝑋𝑖𝑛𝑖=1 is a minimal sufficient statistic for 𝜃2. (9)

ii) Show whether or not ∑ 𝑋𝑖𝑛𝑖=1 is a pivotal quantity. Give two reasons why or

why not it is a pivotal quantity. (5)

iii) What is the distribution of ∑ 𝑋𝑖𝑛𝑖=1 .